On degeneracy loci of equivariant bi-vector fields on a smooth toric variety

We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\le 2k$ is not empty and has at least a component of dimension $\ge 2k+1$, for all integers $k>0$ such that $2k<n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.